\(\int \cos ^4(c+d x) (a+b \tan ^2(c+d x))^2 \, dx\) [448]
Optimal result
Integrand size = 23, antiderivative size = 87 \[
\int \cos ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {1}{8} \left (3 a^2+2 a b+3 b^2\right ) x+\frac {3 \left (a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(a-b) \cos ^3(c+d x) \sin (c+d x) \left (a+b \tan ^2(c+d x)\right )}{4 d}
\]
[Out]
1/8*(3*a^2+2*a*b+3*b^2)*x+3/8*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)/d+1/4*(a-b)*cos(d*x+c)^3*sin(d*x+c)*(a+b*tan(d*x
+c)^2)/d
Rubi [A] (verified)
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3756, 424, 393, 209}
\[
\int \cos ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {3 \left (a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x \left (3 a^2+2 a b+3 b^2\right )+\frac {(a-b) \sin (c+d x) \cos ^3(c+d x) \left (a+b \tan ^2(c+d x)\right )}{4 d}
\]
[In]
Int[Cos[c + d*x]^4*(a + b*Tan[c + d*x]^2)^2,x]
[Out]
((3*a^2 + 2*a*b + 3*b^2)*x)/8 + (3*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((a - b)*Cos[c + d*x]^3*Sin[
c + d*x]*(a + b*Tan[c + d*x]^2))/(4*d)
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 393
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])
Rule 424
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a*d - c*b)*x*(a + b*x^n)^(
p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 1))), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]
Rule 3756
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{\left (1+x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {(a-b) \cos ^3(c+d x) \sin (c+d x) \left (a+b \tan ^2(c+d x)\right )}{4 d}+\frac {\text {Subst}\left (\int \frac {a (3 a+b)+b (a+3 b) x^2}{\left (1+x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 d} \\ & = \frac {3 \left (a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(a-b) \cos ^3(c+d x) \sin (c+d x) \left (a+b \tan ^2(c+d x)\right )}{4 d}+\frac {\left (3 a^2+2 a b+3 b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 d} \\ & = \frac {1}{8} \left (3 a^2+2 a b+3 b^2\right ) x+\frac {3 \left (a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {(a-b) \cos ^3(c+d x) \sin (c+d x) \left (a+b \tan ^2(c+d x)\right )}{4 d} \\
\end{align*}
Mathematica [A] (verified)
Time = 1.47 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.75
\[
\int \cos ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {4 \left (3 a^2+2 a b+3 b^2\right ) (c+d x)+8 \left (a^2-b^2\right ) \sin (2 (c+d x))+(a-b)^2 \sin (4 (c+d x))}{32 d}
\]
[In]
Integrate[Cos[c + d*x]^4*(a + b*Tan[c + d*x]^2)^2,x]
[Out]
(4*(3*a^2 + 2*a*b + 3*b^2)*(c + d*x) + 8*(a^2 - b^2)*Sin[2*(c + d*x)] + (a - b)^2*Sin[4*(c + d*x)])/(32*d)
Maple [A] (verified)
Time = 8.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.18
| | |
| method | result | size |
| | |
| risch |
\(\frac {3 x \,a^{2}}{8}+\frac {x a b}{4}+\frac {3 x \,b^{2}}{8}+\frac {\sin \left (4 d x +4 c \right ) a^{2}}{32 d}-\frac {\sin \left (4 d x +4 c \right ) a b}{16 d}+\frac {\sin \left (4 d x +4 c \right ) b^{2}}{32 d}+\frac {\sin \left (2 d x +2 c \right ) a^{2}}{4 d}-\frac {\sin \left (2 d x +2 c \right ) b^{2}}{4 d}\) |
\(103\) |
| derivativedivides |
\(\frac {b^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) |
\(122\) |
| default |
\(\frac {b^{2} \left (-\frac {\left (\sin \left (d x +c \right )^{3}+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{3} \sin \left (d x +c \right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )+a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) |
\(122\) |
| | |
|
|
|
|
[In]
int(cos(d*x+c)^4*(a+b*tan(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
[Out]
3/8*x*a^2+1/4*x*a*b+3/8*x*b^2+1/32/d*sin(4*d*x+4*c)*a^2-1/16/d*sin(4*d*x+4*c)*a*b+1/32/d*sin(4*d*x+4*c)*b^2+1/
4/d*sin(2*d*x+2*c)*a^2-1/4/d*sin(2*d*x+2*c)*b^2
Fricas [A] (verification not implemented)
none
Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.86
\[
\int \cos ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {{\left (3 \, a^{2} + 2 \, a b + 3 \, b^{2}\right )} d x + {\left (2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{2} + 2 \, a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d}
\]
[In]
integrate(cos(d*x+c)^4*(a+b*tan(d*x+c)^2)^2,x, algorithm="fricas")
[Out]
1/8*((3*a^2 + 2*a*b + 3*b^2)*d*x + (2*(a^2 - 2*a*b + b^2)*cos(d*x + c)^3 + (3*a^2 + 2*a*b - 5*b^2)*cos(d*x + c
))*sin(d*x + c))/d
Sympy [F]
\[
\int \cos ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2} \cos ^{4}{\left (c + d x \right )}\, dx
\]
[In]
integrate(cos(d*x+c)**4*(a+b*tan(d*x+c)**2)**2,x)
[Out]
Integral((a + b*tan(c + d*x)**2)**2*cos(c + d*x)**4, x)
Maxima [A] (verification not implemented)
none
Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.11
\[
\int \cos ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\frac {{\left (3 \, a^{2} + 2 \, a b + 3 \, b^{2}\right )} {\left (d x + c\right )} + \frac {{\left (3 \, a^{2} + 2 \, a b - 5 \, b^{2}\right )} \tan \left (d x + c\right )^{3} + {\left (5 \, a^{2} - 2 \, a b - 3 \, b^{2}\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{4} + 2 \, \tan \left (d x + c\right )^{2} + 1}}{8 \, d}
\]
[In]
integrate(cos(d*x+c)^4*(a+b*tan(d*x+c)^2)^2,x, algorithm="maxima")
[Out]
1/8*((3*a^2 + 2*a*b + 3*b^2)*(d*x + c) + ((3*a^2 + 2*a*b - 5*b^2)*tan(d*x + c)^3 + (5*a^2 - 2*a*b - 3*b^2)*tan
(d*x + c))/(tan(d*x + c)^4 + 2*tan(d*x + c)^2 + 1))/d
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 3916 vs. \(2 (81) = 162\).
Time = 22.62 (sec) , antiderivative size = 3916, normalized size of antiderivative = 45.01
\[
\int \cos ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=\text {Too large to display}
\]
[In]
integrate(cos(d*x+c)^4*(a+b*tan(d*x+c)^2)^2,x, algorithm="giac")
[Out]
1/32*(3*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*
tan(c))*tan(d*x)^4*tan(c)^4 - 5*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*ta
n(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^4 + 12*a^2*d*x*tan(d*x)^4*tan(c)^4 + 8*a*b*d*x*tan(d*x)^4*ta
n(c)^4 + 12*b^2*d*x*tan(d*x)^4*tan(c)^4 + 3*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x)
- 2*tan(c))*tan(d*x)^4*tan(c)^4 - 5*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*ta
n(c))*tan(d*x)^4*tan(c)^4 + 6*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(
c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 - 10*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2
*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 + 6*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^
2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^4 - 10*pi*b^2
*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*
x)^2*tan(c)^4 + 6*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^4 - 10*b^2*arctan((t
an(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^4 - 6*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan
(c) + 1))*tan(d*x)^4*tan(c)^4 + 10*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^4
+ 24*a^2*d*x*tan(d*x)^4*tan(c)^2 + 16*a*b*d*x*tan(d*x)^4*tan(c)^2 + 24*b^2*d*x*tan(d*x)^4*tan(c)^2 + 6*pi*a*b*
sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 - 10*pi*b^2*sgn(-2
*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4*tan(c)^2 + 24*a^2*d*x*tan(d*x)^2*
tan(c)^4 + 16*a*b*d*x*tan(d*x)^2*tan(c)^4 + 24*b^2*d*x*tan(d*x)^2*tan(c)^4 + 6*pi*a*b*sgn(-2*tan(d*x)^2*tan(c)
+ 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^4 - 10*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*t
an(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^4 + 3*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2
*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4 - 5*pi*b^2*sgn(2*tan(d*x)^2*tan(c
)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4 + 12*pi*a*b*sgn(2*
tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*ta
n(c)^2 - 20*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x)
- 2*tan(c))*tan(d*x)^2*tan(c)^2 + 12*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^2
- 20*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4*tan(c)^2 - 12*a*b*arctan(-(tan(d*x) - t
an(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4*tan(c)^2 + 20*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*
tan(d*x)^4*tan(c)^2 - 20*a^2*tan(d*x)^4*tan(c)^3 + 8*a*b*tan(d*x)^4*tan(c)^3 + 12*b^2*tan(d*x)^4*tan(c)^3 + 3*
pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*
tan(c)^4 - 5*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x)
- 2*tan(c))*tan(c)^4 + 12*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^4 - 20*b^2*
arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^4 - 12*a*b*arctan(-(tan(d*x) - tan(c))/(ta
n(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^4 + 20*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2
*tan(c)^4 - 20*a^2*tan(d*x)^3*tan(c)^4 + 8*a*b*tan(d*x)^3*tan(c)^4 + 12*b^2*tan(d*x)^3*tan(c)^4 + 12*a^2*d*x*t
an(d*x)^4 + 8*a*b*d*x*tan(d*x)^4 + 12*b^2*d*x*tan(d*x)^4 + 3*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(
c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^4 - 5*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x
) - 2*tan(c))*tan(d*x)^4 + 48*a^2*d*x*tan(d*x)^2*tan(c)^2 + 32*a*b*d*x*tan(d*x)^2*tan(c)^2 + 48*b^2*d*x*tan(d*
x)^2*tan(c)^2 + 12*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*t
an(c)^2 - 20*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2*tan(c)^
2 + 12*a^2*d*x*tan(c)^4 + 8*a*b*d*x*tan(c)^4 + 12*b^2*d*x*tan(c)^4 + 3*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan
(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^4 - 5*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2
*tan(d*x) - 2*tan(c))*tan(c)^4 + 6*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)
*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2 - 10*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan
(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(d*x)^2 + 6*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan
(c) - 1))*tan(d*x)^4 - 10*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^4 - 6*a*b*arctan(-(ta
n(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^4 + 10*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))
*tan(d*x)^4 - 12*a^2*tan(d*x)^4*tan(c) - 8*a*b*tan(d*x)^4*tan(c) + 20*b^2*tan(d*x)^4*tan(c) + 6*pi*a*b*sgn(2*t
an(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^2 - 10*
pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*
tan(c)^2 + 24*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^2 - 40*b^2*arctan((tan(d
*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2*tan(c)^2 - 24*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c)
+ 1))*tan(d*x)^2*tan(c)^2 + 40*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2*tan(c)^2 + 2
4*a^2*tan(d*x)^3*tan(c)^2 - 48*a*b*tan(d*x)^3*tan(c)^2 + 24*b^2*tan(d*x)^3*tan(c)^2 + 24*a^2*tan(d*x)^2*tan(c)
^3 - 48*a*b*tan(d*x)^2*tan(c)^3 + 24*b^2*tan(d*x)^2*tan(c)^3 + 6*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(
c) - 1))*tan(c)^4 - 10*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(c)^4 - 6*a*b*arctan(-(tan(d*x
) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(c)^4 + 10*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(c)
^4 - 12*a^2*tan(d*x)*tan(c)^4 - 8*a*b*tan(d*x)*tan(c)^4 + 20*b^2*tan(d*x)*tan(c)^4 + 24*a^2*d*x*tan(d*x)^2 + 1
6*a*b*d*x*tan(d*x)^2 + 24*b^2*d*x*tan(d*x)^2 + 6*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan
(d*x) - 2*tan(c))*tan(d*x)^2 - 10*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c
))*tan(d*x)^2 + 24*a^2*d*x*tan(c)^2 + 16*a*b*d*x*tan(c)^2 + 24*b^2*d*x*tan(c)^2 + 6*pi*a*b*sgn(-2*tan(d*x)^2*t
an(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^2 - 10*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x
)*tan(c)^2 + 2*tan(d*x) - 2*tan(c))*tan(c)^2 + 3*pi*a*b*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*tan(c
) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) - 5*pi*b^2*sgn(2*tan(d*x)^2*tan(c)^2 - 2)*sgn(-2*tan(d*x)^2*t
an(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) + 12*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1)
)*tan(d*x)^2 - 20*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(d*x)^2 - 12*a*b*arctan(-(tan(d*x)
- tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*x)^2 + 20*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(d*
x)^2 + 12*a^2*tan(d*x)^3 + 8*a*b*tan(d*x)^3 - 20*b^2*tan(d*x)^3 - 24*a^2*tan(d*x)^2*tan(c) + 48*a*b*tan(d*x)^2
*tan(c) - 24*b^2*tan(d*x)^2*tan(c) + 12*a*b*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(c)^2 - 20*b^
2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1))*tan(c)^2 - 12*a*b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*ta
n(c) + 1))*tan(c)^2 + 20*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1))*tan(c)^2 - 24*a^2*tan(d*x)*tan
(c)^2 + 48*a*b*tan(d*x)*tan(c)^2 - 24*b^2*tan(d*x)*tan(c)^2 + 12*a^2*tan(c)^3 + 8*a*b*tan(c)^3 - 20*b^2*tan(c)
^3 + 12*a^2*d*x + 8*a*b*d*x + 12*b^2*d*x + 3*pi*a*b*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x
) - 2*tan(c)) - 5*pi*b^2*sgn(-2*tan(d*x)^2*tan(c) + 2*tan(d*x)*tan(c)^2 + 2*tan(d*x) - 2*tan(c)) + 6*a*b*arcta
n((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1)) - 10*b^2*arctan((tan(d*x) + tan(c))/(tan(d*x)*tan(c) - 1)) - 6*a*
b*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1)) + 10*b^2*arctan(-(tan(d*x) - tan(c))/(tan(d*x)*tan(c) + 1
)) + 20*a^2*tan(d*x) - 8*a*b*tan(d*x) - 12*b^2*tan(d*x) + 20*a^2*tan(c) - 8*a*b*tan(c) - 12*b^2*tan(c))/(d*tan
(d*x)^4*tan(c)^4 + 2*d*tan(d*x)^4*tan(c)^2 + 2*d*tan(d*x)^2*tan(c)^4 + d*tan(d*x)^4 + 4*d*tan(d*x)^2*tan(c)^2
+ d*tan(c)^4 + 2*d*tan(d*x)^2 + 2*d*tan(c)^2 + d)
Mupad [B] (verification not implemented)
Time = 12.34 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.07
\[
\int \cos ^4(c+d x) \left (a+b \tan ^2(c+d x)\right )^2 \, dx=x\,\left (\frac {3\,a^2}{8}+\frac {a\,b}{4}+\frac {3\,b^2}{8}\right )-\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-\frac {5\,a^2}{8}+\frac {a\,b}{4}+\frac {3\,b^2}{8}\right )-{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {3\,a^2}{8}+\frac {a\,b}{4}-\frac {5\,b^2}{8}\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4+2\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}
\]
[In]
int(cos(c + d*x)^4*(a + b*tan(c + d*x)^2)^2,x)
[Out]
x*((a*b)/4 + (3*a^2)/8 + (3*b^2)/8) - (tan(c + d*x)*((a*b)/4 - (5*a^2)/8 + (3*b^2)/8) - tan(c + d*x)^3*((a*b)/
4 + (3*a^2)/8 - (5*b^2)/8))/(d*(2*tan(c + d*x)^2 + tan(c + d*x)^4 + 1))